Let \(\Vert  \cdot \Vert _{H^p(d)}\)  and \(\Vert  \cdot \Vert _{D_\frac{p}{2}(d)}\)  be the obvious choice of norms for these spaces. As a consequence of the contractivity of \(\iota : D_{\frac{p}{2}} \rightarrow H^p\) , we can deploy the ingenious argument of Helson [1]} (also known as Bonami's lemma because of Lemma 1 in [2]}) to show that the inclusion of \(D_{\frac{p}{2}} (d)\)  in \(H^p (d)\)  is also contractive for all \(d \ge 2\) . Since this technique is considered standard for the experts, we omit the details of the proof.
